Final answer:
The probability density function (PDF) for the percentage of people speaking a language other than English is given by an exponential distribution with a mean of 9.848. The standard deviation is equal to the mean, and the probability P(X < 8) can be calculated using the distribution's cumulative density function. The exponential distribution is apt for this scenario as it can appropriately model the range of percentages across states.
Step-by-step explanation:
The question you've asked pertains to the exponential distribution, which is a type of continuous probability distribution commonly used to model the time between events, or in this case, the percentage of people in a state who speak a language other than English at home. Given the mean μ = 9.848, we can find the probability density function (PDF), standard deviation ( σ ), probability P(X < 8), and other related statistics.
The random variable X, which represents the percentage of persons who speak a language other than English at home in a randomly chosen state, is continuous because it can take on any value within an interval.
The probability density function (PDF) for X, the exponential distribution, is given by:
f(x) = (1/μ) * e^(-x/μ),
The mean (μ) of the distribution is already provided as 9.848, and the standard deviation (σ) of an exponential distribution is equal to the mean, so σ = 9.848.
To find P(X < 8), we integrate the PDF from 0 to 8, resulting in:
P(X < 8) = 1 - e^(-8/9.848).
The exponential distribution is suitable for modeling the percentage of people who speak a language other than English because it can model the probability of a state having a very high or very low percentage, with more states having lower percentages, consistent with the nature of an exponential distribution.